question_answer

If direction of two sides of a triangle are fixed and length of third side is constant and is sliding between these sides, then locus of the orthocenter of the triangle is

A) Circle

B) Ellipse

C) Straight line

D) Hyperbola

Correct Answer: B

Solution :

Let fixed direction be \[OA\] and \[OB\]inclined at a constant angle \[\alpha \]and\[AB=c,\]

Let \[\angle BAO=\theta \,\,BC=c\,\sin \theta \]and \[AC=c\,\cos \theta .\]

\[\therefore \,OC=c\sin \theta .\cot \alpha \]

Equation of line passing through A and perpendicular to \[OB\]is

\[y=-\cot \alpha (x-c\sin \theta \cot \alpha -\cos \theta )\]

And equation of \[BC\]is \[x=c\,\sin \theta .\cot \alpha \]

\[\therefore \]Orthocenter is \[(c\sin \theta .\cot \alpha ,c\,\cos \theta .\cot \alpha )\]

Eliminating \[\theta \] from \[x=c\sin \theta \cot \alpha \]and

\[y=c\cos \theta \cot \alpha \]

\[\Rightarrow \]Required locus is \[{{x}^{2}}+{{y}^{2}}={{c}^{2}}{{\cot }^{2}}\alpha ,\]which the equation of a circle is.